Verifiable Quantum Homomorphic Encryption

Quantum homomorphic encryption (QHE) can allow clients to directly perform quantum computation on encrypted data with the assistance of a remote quantum server. However, existing QHE schemes often overlook the crucial property of verifiability which enables clients to validate the correctness of computation results provided by the server. In this paper, we propose a verifiable QHE scheme based on the universal quantum gate set {H,P,Toffoli}. At first, a specialized gadget is designed to eliminate the errors that may arise during the homomorphic evaluation of non-Clifford Toffoli gates in a non-interactive manner. Furthermore, the designed gadget is versatile and can be seamlessly integrated into an existing QHE scheme that implements quantum gates in another universal quantum gate set {H,T,CNOT} to homomorphically evaluate more quantum gates. Subsequently, a verifiable method is introduced to the QHE scheme based on {H,P,Toffoli} for the client to detect whether the server is honest during the homomorphic computation by employing three types of indistinguishable quantum circuits.